Putnam 1983 B1

Let $v$ be a vertex of a cube $C$ with edges of length $4$ . Let $S$ be the largest sphere that can be inscribed in $C$ . Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$ .